The primary purpose of this paper is to codify a number of independent but interrelated results in linear, multivariable, discrete time systems. In particular, a relationship between forward shift and backward shift representations is established. Conditions for causality in either representation are presented, as are algorithms for obtaining state-space representations of a large class of causal, discrete time systems. The effect on the transfer matrix of non-zero initial conditions is explicitly shown. The finite and infinite poles and zeros of square, invertible systems are defined and related to the lack of row or column ‘ properness ’ of particular representations. The role of the interactor and ‘ inverse interactor ’ in zero determination is also discussed.